NAG Library Routine Document

Integer, Intent (In)	::	m, n, k, lda, lwork
Integer, Intent (Out)	::	info
Real (Kind=nag_wp), Intent (In)	::	tau(*)
Real (Kind=nag_wp), Intent (Inout)	::	a(lda,*)
Real (Kind=nag_wp), Intent (Out)	::	work(max(1,lwork))

C Header Interface

#include <nagmk26.h>

void	f08ajf_ (const Integer m, const Integer n, const Integer k, double a[], const Integer lda, const double tau[], double work[], const Integer lwork, Integer info)

The routine may be called by its LAPACK name dorglq.

3

Description

f08ajf (dorglq) is intended to be used after a call to f08ahf (dgelqf), which performs an

L Q

factorization of a real matrix

A

. The orthogonal matrix

Q

is represented as a product of elementary reflectors.

This routine may be used to generate

Q

explicitly as a square matrix, or to form only its leading rows.

Usually

Q

is determined from the

L Q

factorization of a

p

n

matrix

A

with

p \leq n

. The whole of

Q

may be computed by:

Call dorglq(n,n,p,a,lda,tau,work,lwork,info)

(note that the array a must have at least

n

rows) or its leading

p

rows by:

Call dorglq(p,n,p,a,lda,tau,work,lwork,info)

The rows of

Q

returned by the last call form an orthonormal basis for the space spanned by the rows of

A

; thus f08ahf (dgelqf) followed by f08ajf (dorglq) can be used to orthogonalize the rows of

A

The information returned by the

L Q

factorization routines also yields the

L Q

factorization of the leading

k

rows of

A

, where

k < p

. The orthogonal matrix arising from this factorization can be computed by:

Call dorglq(n,n,k,a,lda,tau,work,lwork,info)

or its leading

k

rows by:

Call dorglq(k,n,k,a,lda,tau,work,lwork,info)

4

References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

5

Arguments

1: $m$ – IntegerInput: On entry: $m$ , the number of rows of the matrix $Q$ .

Constraint: $m \geq 0$ .
2: $n$ – IntegerInput: On entry: $n$ , the number of columns of the matrix $Q$ .

Constraint: $n \geq m$ .
3: $k$ – IntegerInput: On entry: $k$ , the number of elementary reflectors whose product defines the matrix $Q$ .

Constraint: $m \geq k \geq 0$ .
4: $a (lda *)$ – Real (Kind=nag_wp) arrayInput/Output: Note: the second dimension of the array a must be at least $\max (1, n)$ .

On entry: details of the vectors which define the elementary reflectors, as returned by f08ahf (dgelqf).

On exit: the $m$ by $n$ matrix $Q$ .
5: $lda$ – IntegerInput: On entry: the first dimension of the array a as declared in the (sub)program from which f08ajf (dorglq) is called.

Constraint: $lda \geq \max (1, m)$ .
6: $tau (*)$ – Real (Kind=nag_wp) arrayInput: Note: the dimension of the array tau must be at least $\max (1, k)$ .

On entry: further details of the elementary reflectors, as returned by f08ahf (dgelqf).
7: $work (\max (1, lwork))$ – Real (Kind=nag_wp) arrayWorkspace: On exit: if $info = 0$ , $work (1)$ contains the minimum value of lwork required for optimal performance.
8: $lwork$ – IntegerInput: On entry: the dimension of the array work as declared in the (sub)program from which f08ajf (dorglq) is called.
If $lwork = - 1$ , a workspace query is assumed; the routine only calculates the optimal size of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued.

Suggested value: for optimal performance, $lwork \geq m \times nb$ , where $nb$ is the optimal block size.

Constraint: $lwork \geq \max (1, m)$ or $lwork = - 1$ .
9: $info$ – IntegerOutput: On exit: $info = 0$ unless the routine detects an error (see Section 6).

6

Error Indicators and Warnings

$info < 0$: If $info = - i$ , argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

7

Accuracy

The computed matrix

Q

differs from an exactly orthogonal matrix by a matrix

E

such that

{‖E‖}_{2} = O (ε),

where

ε

is the machine precision.

8

Parallelism and Performance

f08ajf (dorglq) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9

Further Comments

The total number of floating-point operations is approximately

4 m n k - 2 (m + n) k^{2} + \frac{4}{3} k^{3}

; when

m = k

, the number is approximately

\frac{2}{3} m^{2} (3 n - m)

The complex analogue of this routine is f08awf (zunglq).

10

Example

This example forms the leading

4

rows of the orthogonal matrix

Q

from the

L Q

factorization of the matrix

A

, where

A = (\begin{array}{r} - 5.42 & 3.28 & - 3.68 & 0.27 & 2.06 & 0.46 \\ - 1.65 & - 3.40 & - 3.20 & - 1.03 & - 4.06 & - 0.01 \\ - 0.37 & 2.35 & 1.90 & 4.31 & - 1.76 & 1.13 \\ - 3.15 & - 0.11 & 1.99 & - 2.70 & 0.26 & 4.50 \end{array}) .

The rows of

Q

form an orthonormal basis for the space spanned by the rows of

A

NAG Library Routine Document

f08ajf (dorglq)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1

Purpose

2

Specification